For two-period crossover trials where the residual carryover can only exist in the presence of treatment effect, Willan (1988, Biometrics 44, 211-218) recommended use of the maximum test statistic that chooses the analysis with the larger test statistic corresponding to the parallel analysis and the crossover analysis. We construct two alternative two-stage test procedures, based on Grizzle's approach, that maintain the actual type I error rate at the desirable level. The power, accuracy, and precision of the analysis based on the modified two-stage procedures are compared to those based on the parallel analysis, crossover analysis, and the maximum test statistic.